G02JDF (PDF version)
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NAG Library Manual

NAG Library Routine Document

G02JDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

G02JDF fits a multi-level linear mixed effects regression model using restricted maximum likelihood (REML). Prior to calling G02JDF the initialization routine G02JCF must be called.

2  Specification

SUBROUTINE G02JDF ( LVPR, VPR, NVPR, GAMMA, EFFN, RNKX, NCOV, LNLIKE, LB, ID, LDID, B, SE, CZZ, LDCZZ, CXX, LDCXX, CXZ, LDCXZ, RCOMM, ICOMM, IOPT, LIOPT, ROPT, LROPT, IFAIL)
INTEGER  LVPR, VPR(LVPR), NVPR, EFFN, RNKX, NCOV, LB, ID(LDID,LB), LDID, LDCZZ, LDCXX, LDCXZ, ICOMM(*), IOPT(LIOPT), LIOPT, LROPT, IFAIL
REAL (KIND=nag_wp)  GAMMA(NVPR+1), LNLIKE, B(LB), SE(LB), CZZ(LDCZZ,*), CXX(LDCXX,*), CXZ(LDCXZ,*), RCOMM(*), ROPT(LROPT)

3  Description

G02JDF fits a model of the form:
y=Xβ+Zν+ε  
where y is a vector of n observations on the dependent variable,
X is a known n by p design matrix for the fixed independent variables,
β is a vector of length p of unknown fixed effects,
Z is a known n by q design matrix for the random independent variables,
ν is a vector of length q of unknown random effects,
and ε is a vector of length n of unknown random errors.
Both ν  and ε  are assumed to have a Gaussian distribution with expectation zero and variance/covariance matrix defined by
Var ν ε = G 0 0 R  
where R= σ R 2 I , I  is the n×n  identity matrix and G  is a diagonal matrix. It is assumed that the random variables, Z , can be subdivided into g q  groups with each group being identically distributed with expectation zero and variance σi2 . The diagonal elements of matrix G  therefore take one of the values σi2 : i=1,2,,g , depending on which group the associated random variable belongs to.
The model therefore contains three sets of unknowns: the fixed effects β, the random effects ν  and a vector of g+1  variance components γ , where γ = σ12,σ22,, σ g-1 2 ,σg2,σR2 . Rather than working directly with γ , G02JDF uses an iterative process to estimate γ* = σ12 / σR2 , σ22 / σR2 ,, σg-12 / σR2 , σg2 / σR2 ,1 . Due to the iterative nature of the estimation a set of initial values, γ0 , for γ*  is required. G02JDF allows these initial values either to be supplied by you or calculated from the data using the minimum variance quadratic unbiased estimators (MIVQUE0) suggested by Rao (1972).
G02JDF fits the model by maximizing the restricted log-likelihood function:
-2 l R = log V + n-p log rT V-1 r + log XT V-1 X + n-p 1+ log 2 π / n-p  
where
V = ZG ZT + R,   r=y-Xb   and   b = XT V-1 X -1 XT V-1 y .  
Once the final estimates for γ *  have been obtained, the value of σR2  is given by
σR2 = rT V-1 r / n - p .  
Case weights, Wc , can be incorporated into the model by replacing XTX  and ZTZ  with XTWcX  and ZTWcZ  respectively, for a diagonal weight matrix Wc .
The log-likelihood, lR, is calculated using the sweep algorithm detailed in Wolfinger et al. (1994).

4  References

Goodnight J H (1979) A tutorial on the SWEEP operator The American Statistician 33(3) 149–158
Harville D A (1977) Maximum likelihood approaches to variance component estimation and to related problems JASA 72 320–340
Rao C R (1972) Estimation of variance and covariance components in a linear model J. Am. Stat. Assoc. 67 112–115
Stroup W W (1989) Predictable functions and prediction space in the mixed model procedure Applications of Mixed Models in Agriculture and Related Disciplines Southern Cooperative Series Bulletin No. 343 39–48
Wolfinger R, Tobias R and Sall J (1994) Computing Gaussian likelihoods and their derivatives for general linear mixed models SIAM Sci. Statist. Comput. 15 1294–1310

5  Arguments

Note: prior to calling G02JDF the initialization routine G02JCF must be called, therefore this documention should be read in conjunction with the document for G02JCF.
In particular some argument names and conventions described in that document are also relevant here, but their definition has not been repeated. Specifically, RNDM, WEIGHT, N, NFF, NRF, NLSV, LEVELS, FIXED, DAT, LICOMM and LRCOMM should be interpreted identically in both routines.
1:     LVPR – INTEGERInput
On entry: the sum of the number of random parameters and the random intercept flags specified in the call to G02JCF.
Constraint: LVPR=iRNDM1,i+RNDM2,i.
2:     VPRLVPR – INTEGER arrayInput
On entry: a vector of flags indicating the mapping between the random variables specified in RNDM and the variance components, σi2. See Section 9 for more details.
Constraint: 1VPRiNVPR, for i=1,2,,LVPR.
3:     NVPR – INTEGERInput
On entry: g, the number of variance components being estimated (excluding the overall variance, σR2).
Constraint: 1NVPRLVPR .
4:     GAMMANVPR+1 – REAL (KIND=nag_wp) arrayInput/Output
On entry: holds the initial values of the variance components, γ0 , with GAMMAi the initial value for σi2/σR2, for i=1,2,,NVPR.
If GAMMA1=-1.0, the remaining elements of GAMMA are ignored and the initial values for the variance components are estimated from the data using MIVQUE0.
On exit: GAMMAi, for i=1,2,,NVPR, holds the final estimate of σi2 and GAMMANVPR+1 holds the final estimate for σR2.
Constraint: GAMMA1=-1.0 ​ or ​ GAMMAi0.0, for i=1,2,,g.
5:     EFFN – INTEGEROutput
On exit: effective number of observations. If there are no weights (i.e., WEIGHT='U'), or all weights are nonzero, then EFFN=N.
6:     RNKX – INTEGEROutput
On exit: the rank of the design matrix, X, for the fixed effects.
7:     NCOV – INTEGEROutput
On exit: number of variance components not estimated to be zero. If none of the variance components are estimated to be zero, then NCOV=NVPR.
8:     LNLIKE – REAL (KIND=nag_wp)Output
On exit: - 2 lR γ^  where lR  is the log of the restricted maximum likelihood calculated at γ^ , the estimated variance components returned in GAMMA.
9:     LB – INTEGERInput
On entry: the dimension of the arrays B and SE and the second dimension of the array ID as declared in the (sub)program from which G02JDF is called.
Constraint: LBNFF+NRF×NLSV.
10:   IDLDIDLB – INTEGER arrayOutput
On exit: an array describing the parameter estimates returned in B. The first NLSV×NRF columns of ID describe the parameter estimates for the random effects and the last NFF columns the parameter estimates for the fixed effects.
The example program for this routine includes a demonstration of decoding the parameter estimates given in B using information from ID.
For fixed effects:
  • for l=NRF×NLSV+1 ,, NRF×NLSV+NFF  
    • if Bl contains the parameter estimate for the intercept then
      ID1l = ID2l = ID3l = 0 ;  
    • if Bl contains the parameter estimate for the ith level of the jth fixed variable, that is the vector of values held in the kth column of DAT when FIXEDj+2=k then
      ID1l=0,  ID2l=j,  ID3l=i;  
    • if the jth variable is continuous or binary, that is LEVELSFIXEDj+2=1, then ID3l=0;
    • any remaining rows of the lth column of ID are set to 0.
For random effects:
  • let
    • NRb denote the number of random variables in the bth random statement, that is NRb=RNDM1,b;
    • Rjb denote the jth random variable from the bth random statement, that is the vector of values held in the kth column of DAT when RNDM2+j,b=k;
    • NSb denote the number of subject variables in the bth random statement, that is NSb=RNDM3+NRb,b;
    • Sjb denote the jth subject variable from the bth random statement, that is the vector of values held in the kth column of DAT when RNDM3+NRb+j,b=k;
    • LSjb denote the number of levels for Sjb, that is LSjb=LEVELSRNDM3+NRb+j,b;
  • then
    • for l=1,2, NRF×NLSV, if Bl contains the parameter estimate for the ith level of Rjb when Skb=sk, for k=1,2,,NSb and 1skLSjb, i.e., sk is a valid value for the kth subject variable, then
      ID1l=b,  ID2l=j,  ID3l=i,  ID3+kl=sk, ​k=1,2,,NSb;  
    • if the parameter being estimated is for the intercept then ID2l=ID3l=0;
    • if the jth variable is continuous, or binary, that is LSjb=1, then ID3l=0;
    • the remaining rows of the lth column of ID are set to 0.
In some situations, certain combinations of variables are never observed. In such circumstances all elements of the lth row of ID are set to -999.
11:   LDID – INTEGERInput
On entry: the first dimension of the array ID as declared in the (sub)program from which G02JDF is called.
Constraint: LDID3+maxj RNDM 3+ RNDM 1,j ,j , i.e., 3+ maximum number of subject variables (see G02JCF).
12:   BLB – REAL (KIND=nag_wp) arrayOutput
On exit: the parameter estimates, with the first NRF×NLSV elements of B containing the parameter estimates for the random effects, ν, and the remaining NFF elements containing the parameter estimates for the fixed effects, β. The order of these estimates are described by the ID argument.
13:   SELB – REAL (KIND=nag_wp) arrayOutput
On exit: the standard errors of the parameter estimates given in B.
14:   CZZLDCZZ* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array CZZ must be at least NRF×NLSV (see G02JCF).
On exit: if NLSV=1, then CZZ holds the lower triangular portion of the matrix 1/ σ 2 ZT R^-1 Z + G^-1 , where R^ and G^ are the estimates of R and G respectively. If NLSV>1 then CZZ holds this matrix in compressed form, with the first NRF columns holding the part of the matrix corresponding to the first level of the overall subject variable, the next NRF columns the part corresponding to the second level of the overall subject variable etc.
15:   LDCZZ – INTEGERInput
On entry: the first dimension of the array CZZ as declared in the (sub)program from which G02JDF is called.
Constraint: LDCZZNRF.
16:   CXXLDCXX* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array CXX must be at least NFF (see G02JCF).
On exit: CXX holds the lower triangular portion of the matrix 1/ σ2 XT V^-1 X , where V^ is the estimated value of V.
17:   LDCXX – INTEGERInput
On entry: the first dimension of the array CXX as declared in the (sub)program from which G02JDF is called.
Constraint: LDCXXNFF.
18:   CXZLDCXZ* – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array CXZ must be at least NLSV×NRF (see G02JCF).
On exit: if NLSV=1, then CXZ holds the matrix 1/σ2 XT V^-1 Z G^ , where V^ and G^ are the estimates of V and G respectively. If NLSV>1 then CXZ holds this matrix in compressed form, with the first NRF columns holding the part of the matrix corresponding to the first level of the overall subject variable, the next NRF columns the part corresponding to the second level of the overall subject variable etc.
19:   LDCXZ – INTEGERInput
On entry: the first dimension of the array CXZ as declared in the (sub)program from which G02JDF is called.
Constraint: LDCXZNFF.
20:   RCOMM* – REAL (KIND=nag_wp) arrayCommunication Array
Note: the dimension of the array RCOMM must be at least LRCOMM (see G02JCF).
On entry: communication array initialized by a call to G02JCF.
21:   ICOMM* – INTEGER arrayCommunication Array
Note: the dimension of the array ICOMM must be at least LICOMM (see G02JCF).
On entry: communication array initialized by a call to G02JCF.
22:   IOPTLIOPT – INTEGER arrayInput
On entry: optional parameters passed to the optimization routine.
By default G02JDF fits the specified model using a modified Newton optimization algorithm as implemented in E04LBF. In some cases, where the calculation of the derivatives is computationally expensive it may be more efficient to use a sequential QP algorithm. The sequential QP algorithm as implemented in E04UCA can be chosen by setting IOPT5=1. If LIOPT<5 or IOPT51 then E04LBF will be used.
Different optional parameters are available depending on the optimization routine used. In all cases, using a value of -1 will cause the default value to be used. In addition only the first LIOPT values of IOPT are used, so for example, if only the first element of IOPT needs changing and default values for all other optional parameters are sufficient LIOPT can be set to 1.
E04LBF is being used.


i


Description
Equivalent
E04LBF
argument


Default Value
1 Number of iterations MAXCAL 1000
2 Unit number for monitoring information n/a As returned by X04ABF
3 Print optional parameters (1= print) n/a -1 (no printing performed)
4 Frequency that monitoring information is printed IPRINT -1
5 Optimizer used n/a n/a
If requested, monitoring information is displayed in a similar format to that given by E04LBF.
E04UCA is being used.


i


Description
Equivalent
E04UCA
argument


Default Value
1 Number of iterations Major Iteration Limit max50,3×NVPR
2 Unit number for monitoring information n/a As returned by X04ABF
3 Print optional parameters (1= print, otherwise no print) List/Nolist -1 (no printing performed)
4 Frequency that monitoring information is printed Major Print Level 0
5 Optimizer used n/a n/a
6 Number of minor iterations Minor Iteration Limit max50,3×NVPR
7 Frequency that additional monitoring information is printed Minor Print Level 0
If LIOPT0 then default values are used for all optional parameters and IOPT is not referenced.
23:   LIOPT – INTEGERInput
On entry: length of the options array IOPT.
24:   ROPTLROPT – REAL (KIND=nag_wp) arrayInput
On entry: optional parameters passed to the optimization routine.
Different optional parameters are available depending on the optimization routine used. In all cases, using a value of -1.0 will cause the default value to be used. In addition only the first LROPT values of ROPT are used, so for example, if only the first element of ROPT needs changing and default values for all other optional parameters are sufficient LROPT can be set to 1.
E04LBF is being used.


i


Description
Equivalent
E04LBF
argument


Default Value
1 Sweep tolerance n/a maxeps,eps×maxizzii
2 Lower bound for γ* n/a eps/100
3 Upper bound for γ* n/a 1020
4 Accuracy of linear minimizations ETA 0.9
5 Accuracy to which solution is required XTOL 0.0
6 Initial distance from solution STEPMX 100000.0
E04UCA is being used.


i


Description
Equivalent
E04UCA
argument


Default Value
1 Sweep tolerance n/a maxeps,eps×maxizzii
2 Lower bound for γ* n/a eps/100
3 Upper bound for γ* n/a 1020
4 Line search tolerance Line Search Tolerance 0.9
5 Optimality tolerance Optimality Tolerance eps0.72
where eps is the machine precision returned by X02AJF and zzii denotes the i diagonal element of ZTZ.
If LROPT0 then default values are used for all optional parameters and ROPT is not referenced.
25:   LROPT – INTEGERInput
On entry: length of the options array ROPT.
26:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
IFAIL=1
On entry, LVPR=value.
Constraint: LVPRvalue.
IFAIL=2
On entry, VPRvalue=value and NVPR=value.
Constraint: 1VPRiNVPR.
IFAIL=3
On entry, NVPR=value.
Constraint: 1NVPRvalue.
IFAIL=4
On entry, GAMMAvalue=value.
Constraint: GAMMA1=-1.0 or GAMMAi0.0.
IFAIL=9
On entry, LB=value.
Constraint: LBvalue.
IFAIL=11
On entry, LDID=value.
Constraint: LDIDvalue.
IFAIL=15
On entry, LDCZZ=value.
Constraint: LDCZZvalue.
IFAIL=17
On entry, LDCXX=value.
Constraint: LDCXXvalue.
IFAIL=19
On entry, LDCXZ= value.
Constraint: LDCXZvalue.
IFAIL=21
On entry, ICOMM has not been initialized correctly.
IFAIL=32
On entry, at least one value of i, for i=1,2,,NVPR, does not appear in VPR.
IFAIL=101
Optimal solution found, but requested accuracy not achieved.
IFAIL=102
Too many major iterations.
IFAIL=103
Current point cannot be improved upon.
IFAIL=104
At least one negative estimate for GAMMA was obtained. All negative estimates have been set to zero.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

Not applicable.

8  Parallelism and Performance

G02JDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02JDF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The argument VPR gives the mapping between the random variables and the variance components. In most cases VPRi=i, for i=1,2,,iRNDM1,i+RNDM2,i. However, in some cases it might be necessary to associate more than one random variable with a single variance component, for example, when the columns of DAT hold dummy variables.
Consider a dataset with three variables:
DAT= 113.6 214.5 311.1 128.3 227.2 326.1  
where the first column corresponds to a categorical variable with three levels, the next to a categorical variable with two levels and the last column to a continuous variable. So in a call to G02JCF
LEVELS=321  
also assume a model with no fixed effects, no random intercept, no nesting and all three variables being included as random effects, then
FIXED=00; RNDM=30123T.  
Each of the three columns in DAT therefore correspond to a single variable and hence there are three variance components, one for each random variable included in the model, so
VPR=123.  
This is the recommended way of supplying the data to G02JDF, however it is possible to reformat the above dataset by replacing each of the categorical variables with a series of dummy variables, one for each level. The dataset then becomes
DAT= 100103.6 010104.5 001101.1 100018.3 010017.2 001016.1  
where each column only has one level
LEVELS= 111111 .  
Again a model with no fixed effects, no random intercept, no nesting and all variables being included as random effects is required, so
FIXED=00 ; RNDM= 60123456T .  
With the data entered in this manner, the first three columns of DAT correspond to a single variable (the first column of the original dataset) as do the next two columns (the second column of the original dataset). Therefore VPR must reflect this
VPR= 111223 .  
In most situations it is more efficient to supply the data to G02JCF in terms of categorical variables rather than transform them into dummy variables.

10  Example

This example fits a random effects model with three levels of nesting to a simulated dataset with 90 observations and 12 variables.

10.1  Program Text

Program Text (g02jdfe.f90)

10.2  Program Data

Program Data (g02jdfe.d)

10.3  Program Results

Program Results (g02jdfe.r)


G02JDF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016